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Identifying Active Sites of the Water-Gas Shift Reaction over Titania Supported Platinum Catalysts under Uncertainty Eric Walker, Donald Mitchell, Gabriel A. Terejanu, and Andreas Heyden ACS Catal., Just Accepted Manuscript • DOI: 10.1021/acscatal.7b03531 • Publication Date (Web): 30 Mar 2018 Downloaded from http://pubs.acs.org on March 31, 2018

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Identifying Active Sites of the Water-Gas Shift Reaction over Titania Supported Platinum Catalysts under Uncertainty Eric A. Walker1, Donald Mitchell2, Gabriel A. Terejanu3,*, Andreas Heyden1,* 1

2

Department of Chemical Engineering, University of South Carolina, 301 Main Street, Columbia, South Carolina 29208, USA

Department of Chemical Engineering, City College of New York, 160 Covenant Avenue, New York, New York, 10031, USA

3

Department of Computer Science and Engineering, University of South Carolina, 301 Main Street, Columbia, South Carolina, 29208, USA

______________________________________ *Corresponding author contact: email: [email protected] 803-777-5025, [email protected] 803-777-5872

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ABSTRACT A comprehensive uncertainty quantification framework has been developed for integrating computational and experimental kinetic data and to identify active sites and reaction mechanisms in catalysis. Three hypotheses regarding the active site for the water-gas shift reaction on Pt/TiO2 catalysts – Pt(111), an edge interface site, and a corner interface site – are tested against experimental kinetic data from three research groups. Uncertainties associated with DFT calculations and model errors of microkinetic models of the active sites are informed and verified using Bayesian inference and predictive validation. Significant evidence is found for the role of the oxide support in the mechanism. Positive evidence is found in support of the edge interface active site over the corner interface site. For the edge interface site, the CO-promoted redox mechanism is found to be the dominant pathway and only at temperatures above 573 K does the classical redox mechanism contribute significantly to the overall rate. At all reaction conditions, water and surface O-H bond dissociation steps at the Pt/TiO2 interface are the main rate controlling steps. KEYWORDS: Active site, Water-gas shift, Density functional theory, Multiscale, Uncertainty, Bayesian


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1. INTRODUCTION Key bottlenecks in the rational design of novel heterogeneous catalysts are our limited ability (i) to integrate experimental, kinetic data with computational, first principles models and (ii) to identify the relevant active sites on the catalyst. In this paper, a framework for overcoming these bottlenecks is presented and applied to the water-gas shift reaction (WGS: 𝐶𝐶𝐶𝐶 + 𝐻𝐻2 𝑂𝑂 ⇌

𝐶𝐶𝑂𝑂2 + 𝐻𝐻2 ) over Pt catalysts supported on titania. Considering that kinetic data such as the turnover frequency and its temperature and pressure dependence are some of the most important parameters

characterizing a heterogeneous catalyst, it is these computational and experimental data that we aim to correlate for the identification of active sites. The WGS is the most widely applied reaction in industry for the generation of hydrogen.1-9 Currently, hydrogen is produced from natural gas sources through a process involving high pressure steam-reforming.10 This process produces syngas (𝐶𝐶𝐶𝐶 + 𝐻𝐻2 + 𝐶𝐶𝐶𝐶2), whose 𝐶𝐶𝐶𝐶 and 𝐻𝐻2 concentration can be adjusted with the addition of

water (H2O) by the WGS. At present, there is disagreement in the literature about the active site of the WGS for Pt catalysts on reducible supports such as TiO2. Some have suggested that the Pt phase is the sole active site, corresponding to terrace active sites studied in this work. This metalonly hypothesis rules out the mechanistic involvement of the support. Grabow et al.11 and Stamatakis et al.12,13 have proposed Pt(111) and Pt(211) as the active site, with little effect due to crystal surface structure. It is to be noted though that Grabow et al.11 arrived at good agreement with experiments only after free energies from DFT were adjusted to the data. In contrast, Schneider et al.14 found a high surface CO coverage for the WGS in simulations on Pt(111) and Pd(111) leading to low turnover frequencies (TOF s-1). Also, for Pt(111) and Pd(111) sites, the reaction orders and apparent activation barrier did not match experiments where Pt and Pd nanoparticles were supported by 𝛾𝛾-Al2O3, a support that has previously been believed to be not ACS Paragon Plus Environment

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active for the WGS.14 A number of research groups15-21 have suggested that most likely the interface of the Pt nanoparticle and the reducible support acts as the active site in most conventionally synthesized catalysts for the WGS. Here, it is still unknown if interface corner or edge sites are the most relevant active sites. Finally, Stephanopoulos et al.22,23 have suggested that single Pt atoms are active for the WGS and could be the primary active site at low temperatures. A microkinetic model based on parameters obtained from first principles by Ammal and Heyden24 confirmed the high activity of atomically dispersed cationic platinum on titania supports, but also suggested that at temperatures above 500 K on most conventionally synthesized Pt catalysts the interface of Pt nanoparticles and the oxide support constitutes the most relevant active site. Previously, Heyden et al.1,25,26 have reported computational models for various reaction mechanisms of the WGS on corner and edge interface sites for a Pt8 nanoparticle supported on rutile TiO2(110). They argue that all threedimensional Pt nanoparticles such as Pt8 on TiO2(110) behave similarly, considering that the interface, oxygen vacancy formation energy is converged with respect to the number of Pt atoms for Pt8, such that their results remains valid for various titania supported Pt nanoparticles. For microkinetic models based on parameters obtained from first principles to conclusively identify the active site and reaction mechanism for the WGS over Pt nanoparticles on titania supports in the experiments from various (here three) research groups,16-18 it is necessary to consider all uncertainties and their correlation in the microkinetic models of the various active sites. Here, we pose this problem of identifying (or more properly eliminating) specific active sites as a Bayesian model selection problem among three hypotheses of active sites investigated by DFT and microkinetic modeling, a Pt(111), interface corner, and interface edge active site model (Figure 1). Bayesian model selection refers to a mathematically defined comparison which


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orders models (here reaction network on specific active sites) according to how well they explain experimental data. The uncertainties associated with the DFT calculations, sticking coefficients, and microkinetic models are modeled using probabilities informed by computational DFT data and experimental data of turnover frequency, apparent activation barrier, and reaction orders, i.e., it is assumed that errors related to harmonic transition state theory used to compute elementary rate constants in the microkinetic models are small relative to model errors and errors related to DFT.16-18 While we admit that at higher temperatures anharmonic effects start to become relevant and harmonic transition state theory has been reported to break down,27 the highest temperature in this work is 573 K which is low enough to at least partially justify our assumption of neglecting anharmonic effects. It is noted that Bayesian statistics has grown in popularity recently28-36 due to the availability of sufficient computational power necessary to solve Bayes’ formula (perform a Bayesian inference, i.e., Bayesian model calibration); however, in computational catalysis, Bayesian statistics has previously not been used to calibrate microkinetic models and perform model selection as well as identify dominant catalytic cycles under uncertainty. Current strategies used to obtain the rate constants for complex chemical reaction networks include top-down approaches, which parametrize and estimate the rate constants based on experimental data, and bottom-up approaches, which calculate rate constants from first principles (e.g., DFT calculations and transition state theory). Neither bottom-up or top-down approaches are satisfactory in determining the rate constants. Top-down strategies require large number of parameters to be estimated and they cannot distinguish between various mechanisms. Bottom-up approaches provide insight into the atomic-scale structure of the catalyst; however, the predicted rates almost never agree with the experimental ones for complex chemistries and active sites, partially due to errors in DFT calculations.


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This work combines the bottom-up and the top-down approaches using a comprehensive uncertainty quantification framework that provides a tight integration between theory and experiments. It uses the bottom-up approach to construct prior distributions over the Gibbs free energies that incorporate correlations due to errors in DFT calculations as previously suggested by Vlachos et al.31, and top-down approach to obtain the posterior distribution of the Gibbs free energies by incorporating the experimental data as suggested by Marzouk et al.30

2. METHODS This section introduces the proposed Bayesian framework for identifying active sites in catalysis (or eliminating specific active sites). The Pt(111) model features reactions occurring on the Pt metal only and the other models feature pathways occurring at a three-phase boundary (TPB) of a Pt nanoparticle and a reducible oxide support, TiO2, see Figure 1. We note that while it is true that other sites with higher TOF may exist theoretically, if their TOF is significantly larger than the measured TOF, then they may not exist on the material which was experimentally evaluated, and they do not need to be considered when identifying active sites. Even if uncertainty in results spans orders of magnitude, a probability may be assigned to each model reflecting how well it explains the experimental data. This comparison can provide insight about the active site and reaction mechanism driving a reaction, here the WGS. A first step in model selection is to perform a Bayesian inference (model calibration) for each site model. The posterior distribution 𝑝𝑝(𝜃𝜃|𝐷𝐷, 𝑀𝑀)

corresponding to the parameters of one of the three models, i.e., 𝑀𝑀 = 𝑀𝑀𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 , is obtained in a

Bayesian model calibration using Bayes’ formula. 𝑝𝑝(𝜃𝜃|𝐷𝐷, 𝑀𝑀) =

𝑝𝑝�𝐷𝐷 �𝜃𝜃, 𝑀𝑀 �𝑝𝑝�𝜃𝜃 �𝑀𝑀 � 𝑝𝑝�𝐷𝐷 �𝑀𝑀 �


(1)

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The parameters 𝜃𝜃 (these are not surface coverages) are the corrections to all surface free energies

from DFT, corrections to gas molecule free energies as well as hyperparameters that describe the distribution of other parameters. The posterior joint probability distribution represents the desired estimate of the parameters with quantified uncertainties given the experimental data, 𝐷𝐷, and all

prior information for the corresponding model. The prior information is encoded in the prior, 𝑝𝑝(𝜃𝜃|𝑀𝑀), which contains all the uncertainty settings including correlations and thermodynamics corrections as presented in Walker et al.25

Four flavors of DFT are used to generate prior distributions in free energy in this work as suggested for this system by Walker et al.25 The reason to use four flavors of DFT functionals is to account for several possible approaches for treating the electronic structure of the system. First, generalized gradient approximation (GGA) functionals are used including the Perdew-BurkeErnzerhof (PBE)37 and Revised Perdew-Burke-Ernzerhof (RPBE)38,39 functionals. Both GGA functionals are known to predict quite different adsorption energies particularly for species containing a CO or CO2 backbone.40 Next, the hybrid Heyd-Scuseria-Ernzerhof (HSE)41 functional that includes exact exchange was included in the uncertainty quantification (UQ) in addition to the M06L meta-GGA functional that has been optimized against a broad set of experimental data including activation barrier information.42 We note that our overall procedure is independent of the specific functionals used as long as DFT errors are not significantly underestimated. For adsorption processes, collision theory is used with an uncorrelated sticking coefficient corresponding to a free energy barrier with a mean of 0.075 eV and a standard deviation of 0.075 eV as done in our prior work.25 Overall gas-phase thermodynamics was corrected to NIST data43 in an unbiased manner using a Dirichlet44 probability density function of free energy corrections as was done previously by Walker et al.25 In this way, the thermodynamics correction is uniformly


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spread among the four gas molecules as it is unknown for the four functionals which molecular free energy is more accurately described than the other gas species. 2.1 Likelihood function and model discrepancy The likelihood function 𝑝𝑝(𝐷𝐷|𝜃𝜃, 𝑀𝑀) provides the likelihood of observing the experimental

data 𝐷𝐷 given the particular values of the parameters and the uncertainty in the model and experiment. Each experimental data set 𝐷𝐷 consists of six individual measurements. 𝐷𝐷 = �𝑇𝑇𝑇𝑇𝑇𝑇, 𝛼𝛼𝐶𝐶𝐶𝐶 , 𝛼𝛼𝐻𝐻2 𝑂𝑂 , 𝛼𝛼𝐶𝐶𝑂𝑂2 , 𝛼𝛼𝐻𝐻2 , 𝐸𝐸𝑎𝑎𝑎𝑎𝑎𝑎 �

(2)

Here, TOF is the turnover frequency, 𝛼𝛼𝑖𝑖 is the reaction order of carbon monoxide, water, carbon

dioxide and hydrogen, respectively, and 𝐸𝐸𝑎𝑎𝑎𝑎𝑎𝑎 (𝑒𝑒𝑒𝑒) is apparent activation energy. We note that

apparent activation energies and reaction orders are determined from TOF at various temperatures

and partial pressures. Considering however that the raw TOF data at various conditions are (unfortunately) often not published, we consider the reaction orders and apparent activation energy as an individual measurement. The six individual measurements are assumed to be independent given model parameters. This translates into the following factorization of the likelihood function. 𝑝𝑝(𝐷𝐷|𝜃𝜃, 𝑀𝑀) = 𝑝𝑝(𝑇𝑇𝑇𝑇𝑇𝑇|𝜃𝜃, 𝑀𝑀)𝑝𝑝(𝛼𝛼𝐶𝐶𝐶𝐶 |𝜃𝜃, 𝑀𝑀)𝑝𝑝�𝛼𝛼𝐻𝐻2 𝑂𝑂 �𝜃𝜃, 𝑀𝑀�𝑝𝑝�𝛼𝛼𝐶𝐶𝑂𝑂2 �𝜃𝜃, 𝑀𝑀�𝑝𝑝�𝛼𝛼𝐻𝐻2 �𝜃𝜃, 𝑀𝑀�𝑝𝑝�𝐸𝐸𝑎𝑎𝑎𝑎𝑎𝑎 �𝜃𝜃, 𝑀𝑀�

(3)

Each individual likelihood function is defined by the discrepancy between the model ∗ simulations, e.g., 𝐸𝐸𝑎𝑎𝑎𝑎𝑎𝑎 and experimental data, e.g., 𝐸𝐸𝑎𝑎𝑎𝑎𝑎𝑎 . This discrepancy is due to unaccounted

model errors and unknown experimental errors. Namely, it is assumed that the discrepancy is normally distributed with zero mean and unknown variance, e.g., 𝜎𝜎𝐸𝐸2𝑎𝑎𝑎𝑎𝑎𝑎 .


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∗ 𝐸𝐸𝑎𝑎𝑎𝑎𝑎𝑎 = 𝐸𝐸𝑎𝑎𝑎𝑎𝑝𝑝 + 𝜖𝜖𝐸𝐸𝑎𝑎𝑎𝑎𝑎𝑎

(4)

Therefore, the fully expanded likelihood function for the WGS calibration is

𝑝𝑝(𝐷𝐷|𝜃𝜃, 𝑀𝑀) =

1

2 �2𝜋𝜋𝜎𝜎𝑇𝑇𝑇𝑇𝑇𝑇

1 (𝑙𝑙𝑙𝑙𝑙𝑙10 𝑇𝑇𝑇𝑇𝑇𝑇−𝑙𝑙𝑙𝑙𝑙𝑙10 𝑇𝑇𝑇𝑇𝐹𝐹 ∗ )

exp �− 2

1

2 �2𝜋𝜋𝜎𝜎𝛼𝛼𝐻𝐻 𝑂𝑂 2

1

2 �2𝜋𝜋𝜎𝜎𝛼𝛼𝐻𝐻

2

2 𝜎𝜎𝑇𝑇𝑇𝑇𝑇𝑇

∗

1 �𝛼𝛼𝐻𝐻2 𝑂𝑂 −𝛼𝛼𝐻𝐻2 𝑂𝑂 �

exp �− 2

𝜎𝜎𝛼𝛼2𝐻𝐻 𝑂𝑂 2 ∗

1 �𝛼𝛼𝐻𝐻2 −𝛼𝛼𝐻𝐻2 �

exp �− 2

𝜎𝜎𝛼𝛼2𝐻𝐻

2

2

�

2

� 1

�

1

�2𝜋𝜋𝜎𝜎𝛼𝛼2𝐶𝐶𝐶𝐶 1

2 �2𝜋𝜋𝜎𝜎𝛼𝛼𝐶𝐶𝑂𝑂

2 �2𝜋𝜋𝜎𝜎𝐸𝐸𝑎𝑎𝑎𝑎𝑎𝑎

2

∗ � 1 �𝛼𝛼𝐶𝐶𝐶𝐶 −𝛼𝛼𝐶𝐶𝐶𝐶

exp �− 2

𝜎𝜎𝛼𝛼2𝐶𝐶𝐶𝐶 ∗

1 �𝛼𝛼𝐶𝐶𝐶𝐶 2 −𝛼𝛼𝐶𝐶𝑂𝑂2 �

exp �− 2

𝜎𝜎𝛼𝛼2𝐶𝐶𝑂𝑂

∗ � 1 �𝐸𝐸𝑎𝑎𝑎𝑎𝑎𝑎 −𝐸𝐸𝑎𝑎𝑎𝑎𝑎𝑎

exp �− 2

𝜎𝜎𝐸𝐸2𝑎𝑎𝑎𝑎𝑎𝑎

2

2

�

2

2

�×

�× (5)

2 2 The hyperparameters 𝜎𝜎𝑇𝑇𝑇𝑇𝑇𝑇 , 𝜎𝜎𝛼𝛼2𝐶𝐶𝐶𝐶 , 𝜎𝜎𝛼𝛼2𝐻𝐻2𝑂𝑂 , 𝜎𝜎𝐶𝐶𝑂𝑂 , 𝜎𝜎𝐻𝐻22 , 𝜎𝜎𝐸𝐸2𝑎𝑎𝑎𝑎𝑎𝑎 are calibrated along with DFT 2

and gas molecule corrections. Hyperparameters may be described as parameters which represent

the distribution of other parameters. The standard deviations of discrepancies are given prior inverse gamma probability density functions (pdfs), which allows them to extend to infinity, however with most of the probability concentrated around a prior value (see Section I (b) of the supporting information). Note that in all models, there is a constraint on the parameters such that the activation barrier for any elementary step is guaranteed to be non-negative (otherwise transition state theory would not be valid). When 𝑁𝑁 experimental data sets are available, {𝐷𝐷}𝑖𝑖=1..𝑁𝑁 , (in this case three), it is assumed

that they are independent and identically distributed. Independent and identically distributed

means that the experiments are not correlated with each other and the experiments have the same uncertainty distribution. The likelihood function is, 𝑝𝑝({𝐷𝐷}𝑖𝑖=1..𝑁𝑁 |𝜃𝜃, 𝑀𝑀) = ∏𝑁𝑁 𝑖𝑖=1 𝑝𝑝(𝐷𝐷𝑖𝑖 |𝜃𝜃, 𝑀𝑀)


(6)

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2.2 Bayesian model selection The evidence, 𝑝𝑝(𝐷𝐷|𝑀𝑀), in Bayesian inference (model calibration), Eq. (1), acts as both a

normalization constant as well as a key quantity in comparing candidate models. It is a natural formulation of Occam’s razor, providing an automatic trade-off between goodness-of-fit and

model complexity. After model calibration, the evidence in Eq. (1), 𝑝𝑝(𝐷𝐷|𝑀𝑀) is used in a second Bayesian inference problem to calculate the posterior probability for all candidate models. 𝑝𝑝(𝑀𝑀|𝐷𝐷) =

𝑝𝑝�𝐷𝐷 �𝑀𝑀 �𝑝𝑝(𝑀𝑀) 𝑝𝑝(𝐷𝐷)

(7)

The log-evidence can be written as the difference between the expected log-likelihood of the data and the Kullback-Leibler45,46 (KL) divergence between posterior and prior pdf of model parameters. The expected log-likelihood quantifies how well the model fits the data, and the KL divergence quantifies model complexity. A large divergence between the posterior and prior pdfs suggests over-fitting of experimental data. Therefore, a complex model is penalized, meaning it might not be selected over a simpler model that does not explain the data as well. This explains its parsimonious model selection property related to Occam’s razor. Note, that KL divergence has also been used by Walker et al.25 to determine the distance of two catalytic cycle TOF (s-1) pdf’s divergence from the overall TOF (s-1). Thus, KL divergence served as a formalization to determining the dominant pathway. In the absence of information regarding which model is better at describing the catalytic mechanism, the prior model probabilities in Eq. (7) are set to 𝑝𝑝�𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 � = 𝑝𝑝(𝑀𝑀𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 ) = 1

𝑝𝑝(𝑀𝑀𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ) = 3. Note that in this case, the evidence can be used directly to compare the proposed models, and the strength of model comparison can be determined using Bayes’ factors. Once both


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three-phase boundary models and the Pt(111) model are calibrated using the same data, the evidences, 𝑝𝑝(𝐷𝐷|𝑀𝑀), of each calibration can be divided to produce a Bayes’ factor. We implicitly assume here that one and only one active site dominates the observed reaction behavior. However,

the single site models in this work may be combined to generate multiple site models that become just another set of candidate models to be discriminated by the proposed methodology.

𝐵𝐵𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒/𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 =

𝑝𝑝�𝐷𝐷 �𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 �

𝑝𝑝�𝐷𝐷 �𝑀𝑀𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 �

(8)

To determine the strength of evidence given by Bayes’ factor in favor of one model against the other, we use Jeffreys scale47 as shown in Table 1. A Bayes factor of e.g. 𝐵𝐵𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒/𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = 3.0 means that the data 𝐷𝐷 is three times more likely under model 𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 than 𝑀𝑀𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 . Bayes factors

are an alternative to classical hypothesis testing, and while the values can be used directly as shown previously, it is easier to have a descriptive statement that can be used to interpret them. For example, there is no difference when the Bayes factor is 104 or 1010, both point to the fact that the data very strongly support the null hypothesis (numerator in Eq. 8) over the alternative (denominator in Eq. 8) as suggested by Jeffreys. The supporting information details further complexities in regard to the order of experimental data points used for the Bayesian inference, sampling from the posterior probability density, 𝑝𝑝(𝜃𝜃|𝐷𝐷, 𝑀𝑀) and approximating the model evidence, 𝑝𝑝(𝐷𝐷|𝑀𝑀). Also, all DFT data used for construction of prior distributions for the three active sites are summarized. Finally, lateral

interactions for the interface corner and edge active sites are incorporated explicitly, and for the Pt terrace model, they are included with a linear lateral interaction model based on PBE data as described in section IV in the supporting information.


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3. RESULTS AND DISCUSSION Each probabilistic model consists of a microkinetic model, a probabilistic discrepancy model to account for errors between model predictions and observations, and a prior distribution over the free energies of intermediates, transition states (SI. I (c), gas molecule corrections and model discrepancy parameters (SI. I(d)) . The results corresponding to the model selection are discussed first, followed by the specific findings for the active site. Model selection. Three datasets (D118, D216, D317 – see SI. II) comprising TOF, apparent activation barrier, and reaction order measurements corresponding to different experimental conditions are used to inform, rank, and validate the proposed probabilistic models. The first experimental dataset is used to constrain the initial DFT-based uncertainty for intermediates and transition states along with the distribution that governs gas molecule corrections and model discrepancies via a Bayesian model calibration (SI. I (a)). This informed distribution becomes the prior distribution in the Bayesian model selection where the second dataset is used to rank the models based on their posterior model probability or evidence in the case of equal prior model probabilities. Finally, the third dataset is used to perform predictive validation to assess the consistency between probabilistic model predictions and experimental data (see SI. III for computational details). Given that Bayesian model selection is highly sensitive to the prior distribution,48 all six possible permutations of the three datasets are used to inform, rank and validate to provide robust findings given all available information.49 Table 2 summarizes the evidence and corresponding Bayes factors with respect to the ranking dataset. In all six cases, the evidence for the terrace site is significantly smaller than for the interface corner and edge sites, which results in “very strong evidence” (see Table 1 - Jeffreys scale47) that the terrace site is not the active site among the three.


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Since the terrace site is the only site which does not include the TiO2 support in the mechanism, a first conclusion may be drawn that the oxide support is mechanistically involved in the WGS. The evidences for the edge and corner sites are not sufficient to further discriminate between them. The evidences are highly dependent on the datasets used to inform the prior. In Table 2 - Group III of permutations, informing the prior using data D1 and ranking the models on data D2 results in positive evidence for the corner site, however, when informing the prior using D2 and ranking on D1, we find positive evidence for the edge site. Given that Bayesian calibration does not guarantee consistency between model predictions and experimental data, a posterior predictive check is used to solve this ambiguity in model selection.50 Mahalanobis distance is used as a consistency check to test whether the experimental data set is a possible outcome of the model considering all quantified uncertainties (SI. III). Even though the Bayes factor indicates that the edge is preferred over the corner site, the consistency check results (the Mahalanobis distance larger than the required threshold) corresponding to Table 2 - Group II of permutations suggest that the datasets D1 and D3 are not sufficient to inform the uncertainty in either of the models and the corresponding results should not be used in model predictions49 or to discriminate between the two interface sites. In the case of Group III of permutations, the calibrated model corresponding to the corner site has a favorable Bayes factor as compared with the edge, however it does fail the consistency check for both the calibration and validation datasets. The same situation arises in Table 2 - Group I of permutations. Overall, these consistency checks provide positive evidence that the edge site is a better descriptor of the observed catalytic activity given all the available information in this study.


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Edge active site. Data from two experiments (D1, D2 – Group III) and (D2, D3 – Group I) are used to calibrate the microkinetic model and obtain the posterior predictive distributions for various quantities of interest. Figure 2 depicts the posterior predictive uncertainty for the overall TOF, apparent activation energy (eV) and reaction orders along with the third experimental dataset used for validation purposes (D1 for Group I and D3 for Group III). Compared with the prior predictive uncertainty (based on DFT data only),25 the posterior uncertainty is reduced while capturing the validation experimental data within the bulk of the probability mass. As with the prior predictive uncertainty, the posterior predictive uncertainty indicates that the CO-promoted redox mechanism is dominant, see Figure 3. It is dominant over the classical redox mechanism in Group III and over the formate mechanism in Group I. This agrees with the free energy pathway being lower for the CO-promoted pathway illustrated in Figure 4. The prior mean corresponding to the edge active site for the dominant CO-promoted redox pathway is slightly higher than the PBE values. The gas molecule corrections provide exact thermodynamics and as a result, no uncertainty is associated with the end of the classical redox pathway shown in Figure 4 (Group III). Also, all free energies are referenced with respect to state S01 such that there is no uncertainty associated with this state. The highest free energy transition state within the CO-promoted redox pathway is the oxygen vacancy formation. The uncertainties in DFT energies and molecule corrections induce a probability distribution over the degrees of rate control (DRC)51-55 which are a measure of the rate controlling steps in the reaction network. Figure 5 shows the mean of the degree of rate control corresponding to various transition states. Key rate controlling steps common to Group I and III simulations at reaction conditions corresponding to validation data D1 and D3, respectively, are TS10 (COPt-Vint +Oint + H2O(g)  COPt-2OHint), TS11 (COPt-2OHint + Os  COPt-OHint-Oint-OHs), and TS12


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ACS Catalysis

(COPt-OHint-Oint-OHs + Oint  COPt-OHint-Oint-OHint + Os). All of these elementary reactions belong to the CO-promoted redox pathway and are water and surface O-H bond dissociations at the Pt/TiO2 interface. In addition, for Group I (low temperature conditions D1), TS08 (CO2(Pt-int) + CO(g)  COPt-CO2(int)), which is a CO adsorption step on a small coverage site in the COpromoted redox pathway, becomes partially rate controlling. For Group III (high temperature conditions D3), a surface O-H bond dissociation (TS06: HPt-OHint + *Pt

 2HPt-Oint) and an

oxygen vacancy formation step (TS03: CO2(Pt-int)  *Pt-Vint + CO2(g)) in the classical redox pathway also become partially rate controlling, illustrating that at temperatures of 573 K both reaction mechanisms, the classical redox and the CO-promoted redox pathway, are operational.

4. CONCLUSIONS Computational catalysis suffers from significant uncertainties in its predictions, primarily due to significant uncertainties in DFT energies. Even when DFT results are combined with microkinetic modeling to simulate experiments from first principles, it is often not possible to conclusively determine when a model (consisting of an active site and reaction mechanism) contains the necessary physics and chemistry to describe experimental, kinetic data. Here, a comprehensive uncertainty quantification framework has been developed for integrating computational and experimental, kinetic catalyst data and to identify active sites and reaction mechanisms in catalysis. This framework was applied to the water-gas shift reaction over Pt catalysts supported on titania. Three actives sites, a Pt(111) terrace model, an edge and a corner interface model, are investigated and the most active site is selected. Four qualitatively different DFT functionals are used to evaluate the prior uncertainty for each site. Using corresponding microkinetic models derived from first principles and experimental kinetic data of TOF, reaction orders and apparent activation barrier, a Bayesian calibration is conducted for each active site


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model. The evidence for the edge and corner site is significantly larger than for the terrace site, which suggests that the terrace site is not the active site in the experiments. The edge and corner sites are both at the three-phase boundary of the Pt nanoparticle and the TiO2 support; thus, we conclude that the support plays a mechanistic role. The statements above assume that the reaction network considered for each active site model is sufficiently complete. Posterior predictive checks are used to discriminate between the edge and the corner site. Consistency between model predictions and experimental data is found in favor of the edge site, which can capture both calibration and validation datasets. Given all available information in this study, we conclude that the edge site is the active site for the WGS in the catalysts studied experimentally when compared with the terrace and interface corner site. Even in the presence of uncertainty, the CO-promoted redox mechanism at the edge active site is found to be the dominant reaction mechanism and only at temperatures above 573 K does the classical redox mechanism contribute also significantly to the overall rate. The prediction of degrees of rate control with quantified uncertainties reveals that at all reaction conditions, water and surface O-H bond dissociation steps at the Pt/TiO2 interface are the main rate controlling steps. At low temperatures of 503 K, a CO adsorption step on a small coverage site in the CO-promoted redox pathway also becomes partially rate controlling while at high temperatures of 573 K an interface TiO2 oxygen vacancy formation step in the classical redox pathway becomes partially rate controlling. Overall, we believe that beyond solving what is the active site for the water-gas shift in the experimental datasets of Pt/TiO2 catalysts, the methodology presented in this work is transferrable to other catalysis challenges where determination of the active site is of critical importance.

ASSOCIATED CONTENT


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Supporting Information The Supporting Information is available free of charge on the ACS Publications website at DOI: XXX Details of proposed Bayesian statistics for identifying active sites in catalysis (or eliminating specific active sites) such as the order of data for points for Bayesian inference, Computational approach, Prior construction for intermediate and transition states, Prior construction for gas molecule corrections and model discrepancy, Experimental data, Model validation via posterior predictive check, Terrance active site model, and Lateral interaction model. Notes The authors declare no competing financial interest.

ACKNOWLEDGEMENTS This work was supported by the National Science Foundation under Grant No. CBET1254352. G.A.T has been supported by NSF DMREF-1534260. D.M. acknowledges funding from NSF EEC-1358931. Furthermore, a portion of this research was performed using XSEDE resources provided by the National Institute for Computational Sciences (NICS), San Diego Supercomputer Center, and Texas advanced Computing Center (TACC) under grant number TGCTS090100 and at the U.S. Department of Energy facilities located at the National Energy Research Scientific Computing Center (NERSC).


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Table 1. Jeffreys scale47 for Bayes factors, 𝐵𝐵12 = 𝑝𝑝(𝐷𝐷|𝑀𝑀1 )⁄𝑝𝑝(𝐷𝐷|𝑀𝑀2 ). 𝐵𝐵12 Evidence against 𝑀𝑀2 1-3.2 Not worth more than a bare mention 3.2-10 Positive 10-100 Strong >100 Very Strong


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Table 2. Evidences and squared Mahalanobis distances for corner and edge sites and experimental conditions for the three datasets. Positive evidences as per Jeffreys’ scale47 (Table 1) and squared Mahalanobis distances smaller than 12.59 and marked with bold, correspond to consistencies between model predictions and calibration and validation data at 0.05 significance level (see SI. III). Note that the evidences for the terrace site are at least 6 orders of magnitude smaller than evidences of corner and edge for all six possible cases. As a result, the metal-only terrace site is not active. Shaded data points in the table are used to validate model predictions, see also Figure 2.

Inform

Rank

Corner (C)

Edge (E)

Terrace (T)

Bayes Factor E/C

𝐷𝐷3

5.96×10-3

5.70×10-12

3.96

I

𝐷𝐷2

1.50×10-3

1.34×10-3

5.92×10-9

1.31

𝐷𝐷3

5.52×10-4

3.10×10-4

1.37×10-10

0.56

II

𝐷𝐷1

𝐷𝐷2

1.02×10-3

𝐷𝐷1

6.71×10-4

1.90×10-3

1.50×10-10

2.83

𝐷𝐷2

9.24×10-4

2.19×10-4

1.47×10-36

0.24

𝐷𝐷1

1.66×10-3

5.99×10-3

1.67×10-33

3.61

Datasets

Evidence

Group

III 𝐷𝐷118 𝐷𝐷216 𝐷𝐷317

𝐷𝐷3 𝐷𝐷3 𝐷𝐷1

𝐷𝐷2

Squared Mahalanobis Distance (

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